Boundary controllability of a chain of serially connected Euler-Bernoulli beams with interior masses

Abstract

The aim is to study the boundary controllability of a system modelling the vibrations of a network of N Euler-Bernoulli beams serially connected by (N − 1) vibrating interior point masses. Using the classical Hilbert Uniqueness Method, the control problem is reduced to the obtention of an observability inequality. The solution is then expressed in terms of Fourier series so that one of the sufficient conditions for the observability inequality is that the distance between two consecutive large eigenvalues of the spatial operator involved in this evolution problem is superior to a minimal fixed value. This property called spectral gap holds. It is proved using the exterior matrix method due to W.H. Paulsen. Two more asymptotic estimates involving the eigenfunctions are required for the observability inequality to hold. They are established using an adequate basis.

Spectral analysis of the Schrödinger operator on binary tree-shaped networks and applications

“…This type of structure was studied by Castro and Zuazua in many papers (see [14] and the papers by the same authors cited in [22] as well as one by Castro and Hansen also cited there).…”

“…In another paper (see [22]), Mercier and Régnier used the technique of exterior matrices due to W.H. Paulsen (presented for other purposes in [23]) which D. Mercier had already used in the same type of context in [20].…”

“…See the references by Ali Mehmeti, von Below and Nicaise in [22] as well as [15,16,19] and the references therein. The spectral analysis of such structures has some applications to control or stabilization problems (cf.…”

“…papers by von Below, Nicaise and [19]), qualitative properties of solutions (see papers by von Below cited in [22] for example) etc...…”

“…For interconnected strings (corresponding to a second-order operator on each string), a lot of results have been obtained: the asymptotic behaviour of the eigenvalues (see the references by Ali Mehmeti, von Below and Nicaise in [22] as well as [2]), the relationship between the eigenvalues and algebraic theory (cf. papers by von Below, Nicaise and [19]), qualitative properties of solutions (see papers by von Below cited in [22] for example) etc...…”